arXiv subjects

Jake Levinson

Publications and source records attributed to Jake Levinson.

26 records · Page 2Linked to original sources

arXiv2018

Towards Boij-Söderberg theory for Grassmannians: the case of square matrices

We characterize the cone of GL-equivariant Betti tables of Cohen-Macaulay modules of codimension 1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for `Boij-Söderberg theory for Grassmannians', with the goal of characterizing the cones of GL_k-equivariant Betti tables of modules over the coordinate ring of k x n matrices, and, dually, cohomology tables of vector bundles on the Grassmannian Gr(k, C^n). The proof uses Hall's Theorem on perfect matchings in bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding equivariant free resolutions by applying Weyman's geometric technique to certain graded pure complexes of Eisenbud-Fløystad-Weyman.

math.AC↗

arXiv2017

Shifted tableaux crystals

We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are the (skew) Schur $Q$-functions. We give local axioms for these crystals, which closely resemble the Stembridge axioms for type A. Finally, we give a new criterion for such tableaux to be ballot.

math.CO↗

arXiv2017

A crystal-like structure on shifted tableaux

math.CO↗

arXiv2016

Foundations of Boij-Söderberg Theory for Grassmannians

Boij-Söderberg theory characterizes syzygies of graded modules and sheaves on projective space. This paper continues earlier work with S. Sam, extending the theory to the setting of $GL_k$-equivariant modules and sheaves on Grassmannians. Algebraically, we study modules over a polynomial ring in $k n$ variables, thought of as the entries of a $k \times n$ matrix. We give equivariant analogues of two important features of the ordinary theory: the Herzog-Kühl equations and the pairing between Betti and cohomology tables. As a necessary step, we also extend previous results, concerning the base case of square matrices, to cover complexes other than free resolutions. Our statements specialize to those of ordinary Boij-Söderberg theory when $k=1$. Our proof of the equivariant pairing gives a new proof in the graded setting: it relies on finding perfect matchings on certain graphs associated to Betti tables. Finally, we give preliminary results on $2 \times 3$ matrices, exhibiting certain classes of extremal rays on the cone of Betti tables.

math.AG↗

arXiv2016

Monodromy and K-theory of Schubert curves via generalized jeu de taquin

We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(λ_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $ω$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $ω$ as the monodromy operator. We provide a local algorithm for computing $ω$ without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the $K$-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S(λ_\bullet)$.

math.CO↗

arXiv2015

Monodromy and $K$-theory of Schubert Curves via Generalized Jeu de Taquin

We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(λ_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $ω$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $ω$ as the monodromy operator. We provide a fast, local algorithm for computing $ω$ without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the $K$-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S(λ_\bullet)$.

math.CO↗

arXiv2015

One-dimensional Schubert problems with respect to osculating flags

We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real -- in this case, for zero-dimensional Schubert problems, the solutions are "as real as possible". Recent work by Speyer has extended the theory to the moduli space $\overline{M_{0,r}}$, allowing the points to collide. These give rise to smooth covers of $\overline{M_{0,r}}(\mathbb{R})$, with structure and monodromy described by Young tableaux and jeu de taquin. In this paper, we give analogous results on one-dimensional Schubert problems over $\overline{M_{0,r}}$. Their (real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over $M_{0,r}$, our results show that the real points of the solution curves are smooth. We also find a new identity involving `first-order' K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

math.AG↗

arXiv2014

The n-level densities of low-lying zeros of quadratic Dirichlet L-functions

Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions where the sum of the supports of the Fourier transforms is at most 2, and showed agreement with random matrix theory predictions in this range for n < 4 but only in a restricted range for larger n. We extend these results and show agreement for n < 8, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.

math.NT↗